Closure of Steinberg Fibers and Affine Deligne-Lusztig Varieties

We discuss some connections between the closure (F) over bar of a Steinberg fiber in the wonderful compactification of an adjoint group and the affine Deligne-Lusztig varieties X(w)(1) in the affine flag variety. Among other things, we describe the emptiness/nonemptiness pattern of X(w)(1) if the translation part of w is quasi-regular. As a by-product, we give a new proof of the explicit description of (F) over bar, first obtained in He ["Unipotent variety in the group compactification." Advances in Mathematics 203(2006):109-31]